Question

If the value of $\alpha = {e^{i\dfrac{{8\pi }}{{11}}}}$ then the real value of $\left( {\alpha + {\alpha ^2} + {\alpha ^3} + {\alpha ^4} + {\alpha ^5}} \right)$ equal to
(A) $0$
(B) $1$
(C) $ - \dfrac{1}{2}$
(D) $ - 1$

Answer 1

Hint: We have given a complex exponent number and we have to determine the real value of the given expression. To find out the real value of the given expression, first, we separate the real part of each term by using the Euler’s formula ${e^{i\theta }} = \cos \theta + i\sin \theta $ .
After that we use the standard result of AP that is
$\cos A + \cos \left( {A + B} \right) + \cos \left( {A + 2B} \right) + ..... + \cos \left( {A + \left( {n - 1} \right)B} \right) = \dfrac{{\cos \left( {A + \dfrac{{\left( {n - 1} \right)B}}{2}} \right)\sin \dfrac{{nB}}{2}}}{{\sin \dfrac{B}{2}}}$
After separating the real terms substitute the values of the cosine of the angles and evaluate the result.

Complete step-by-step answer:
Step1: Apply Euler’s formula to determine the value of $\alpha $
We have given the value of $\alpha = {e^{i\dfrac{{8\pi }}{{11}}}}$ . We apply Euler’s formula, we get
\[ \Rightarrow \alpha = \cos \dfrac{{8\pi }}{{11}} + i\sin \dfrac{{8\pi }}{{11}}\] …..(1)
Step2: Find the value of each term in the given expression
The given expression is $\left( {\alpha + {\alpha ^2} + {\alpha ^3} + {\alpha ^4} + {\alpha ^5}} \right)$
The value of ${\alpha ^2}$ is given as \[{\alpha ^2} = {\left( {{e^{i\dfrac{{8\pi }}{{11}}}}} \right)^2}\]
Applying the exponent rule, we get \[{\alpha ^2} = {e^{i\dfrac{{16\pi }}{{11}}}}\]
Applying Euler’s formula, we get \[{\alpha ^2} = \cos \dfrac{{16\pi }}{{11}} + i\sin \dfrac{{16\pi }}{{11}}\] …..(2)
The value of ${\alpha ^2}$ is given as \[{\alpha ^3} = {\left( {{e^{i\dfrac{{8\pi }}{{11}}}}} \right)^3}\]
Applying the exponent rule, we get \[{\alpha ^3} = {e^{i\dfrac{{24\pi }}{{11}}}}\]
Applying Euler’s formula, we get \[{\alpha ^3} = \cos \dfrac{{24\pi }}{{11}} + i\sin \dfrac{{24\pi }}{{11}}\] …..(3)
The value of ${\alpha ^4}$ is given as \[{\alpha ^4} = {\left( {{e^{i\dfrac{{8\pi }}{{11}}}}} \right)^4}\]
Applying the exponent rule, we get \[{\alpha ^4} = {e^{i\dfrac{{32\pi }}{{11}}}}\]
Applying Euler’s formula, we get \[{\alpha ^4} = \cos \dfrac{{32\pi }}{{11}} + i\sin \dfrac{{32\pi }}{{11}}\] …..(4)
The value of ${\alpha ^5}$ is given as \[{\alpha ^5} = {\left( {{e^{i\dfrac{{8\pi }}{{11}}}}} \right)^5}\]
Applying the exponent rule, we get \[{\alpha ^5} = {e^{i\dfrac{{40\pi }}{{11}}}}\]
Applying Euler’s formula, we get \[{\alpha ^5} = \cos \dfrac{{40\pi }}{{11}} + i\sin \dfrac{{40\pi }}{{11}}\] …..(5)
Step3: Add the real part of each term
Now we add the real part of each term in equation (1) to equation (5), we get
$
  \operatorname{Re} \left( {\alpha + {\alpha ^2} + {\alpha ^3} + {\alpha ^4} + {\alpha ^5}} \right) \\
   \Rightarrow \cos \dfrac{{8\pi }}{{11}} + \cos \dfrac{{16\pi }}{{11}} + \cos \dfrac{{24\pi }}{{11}} + \cos \dfrac{{32\pi }}{{11}} + \cos \dfrac{{40\pi }}{{11}} \\
 $
Step 4: Substitute the values
Now substituting the values of each term, we get
$
   \Rightarrow \cos \dfrac{{8\pi }}{{11}} + \cos \dfrac{{16\pi }}{{11}} + \cos \dfrac{{24\pi }}{{11}} + \cos \dfrac{{32\pi }}{{11}} + \cos \dfrac{{40\pi }}{{11}} \\
   \Rightarrow - 0.654 - 0.142 + 0.841 - 0.959 + 0.415 \\
   \Rightarrow - 0.50 \\
 $

So the value of \[\operatorname{Re} \left( {\alpha + {\alpha ^2} + {\alpha ^3} + {\alpha ^4} + {\alpha ^5}} \right) = - \dfrac{1}{2}\].

Note:
To solve such a type of question, separate the real and imaginary part first and then simplify for the real part.
Commit to memory:
Euler’s Formula: ${e^{i\theta }} = \cos \theta + i\sin \theta $
Power law of exponent ${\left( {{a^m}} \right)^n} = {a^{m \times n}}$